All Questions
Tagged with vector-fieldshomework-and-exercises
193 questions
-4votes
2answers
178views
Reason why magnetic field lines never intersect [closed]
Assertion (A): Magnetic field lines around a bar magnet never intersect each other. Reason (R): Magnetic field produced by a bar magnet is a quantity that has both magnitude and direction Is the ...
1vote
0answers
63views
Determine the maximum and minimum acceleration from the velocity field [closed]
If the speed distribution, in m/s, of a flow is given by $v = 2x^3 + 2y^2 - 3z$, then the acceleration of the fluid at the point with coordinates $[2, 1, 5]$, in meters, will be greater than $20, \...
0votes
1answer
102views
Problem in deriving Killing equation
I am studying derivation of Killing equation by Wald (also reading some other literature) but having some problem in understanding the math. Let $\chi ^a$ is killing vector on the horizon $$\chi _{[a} ...
- 595
1vote
1answer
154views
Finding Killing vectors for hyperbolic space [closed]
I want to find the Killing vectors for the hyperbolic space, which is described by the metric \begin{equation} ds^2 = \frac{dx^2 + dy^2}{y^2}. \end{equation} I have found the Killing equations, which ...
5votes
4answers
659views
Vector triple product with $\nabla$ operator
I came across the following expression in several books (especially in plasma physics literature while deriving the magnetic pressure): $$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\...
- 61
0votes
1answer
86views
Method of image charges for ungrounded conductive sphere seems to have charge of $q$ and not $(r/a) q$?
Using the 2d scenario for simplification Vector field of a point charge $q=1$ at (-4.1,0): $$\vec F_1\left( {x,y} \right) = \left(\frac{x+4.1} {{\sqrt{(x+4.1)^2 + (y-0)^2}}^2}\right) \vec e_x + \left(\...
- 217
2votes
2answers
273views
Vector potential of position field
Consider the position vector field $\vec{r}=(x,y,z)^T$. What would be a vector potential $\vec{A}$ for this field? I was thinking of something like $\vec{A}=(yz,zx,-xy)^T$, which gives $$\nabla\times ...
- 1,609
1vote
1answer
121views
Spherical coordinate of a vector when divergence of the vector is zero
$\nabla \cdot \mathbf{\delta u_{perp}} = 0$ where $\mathbf{\delta u_{perp}}$ is a function of both x and y coordinates and perpendicular to z axis. Moreover, $\delta u_{perp}$ along z axis is $0$. I ...
- 31
0votes
1answer
80views
Understanding the derivation of Killing horizon surface gravity
In the book "A Relativist's Toolkit" by Eric Poisson, he explains surface gravity in section 5.2.4 The equation 5.40 says $$ (-t^\mu t_\mu)_{;\alpha} = 2 \kappa t_\alpha \tag{5.40}$$ where $...
- 323
0votes
0answers
98views
Deriving divergence in cylindrical coordinates, using covariant derivatives
Covariant derivatives are normally used to write equations covariantly in curved spaces. But in an exercise, I need to use covariant derivatives to derive Gauss' law: $\nabla \cdot \vec{E} = 4\pi\rho$ ...
- 145
0votes
1answer
64views
Surface Integration [closed]
I've been studying surface integration by myself but I always stuck at the last step. Consider the above question: This is my approach: Calculation of the curl of the given field. Calculation of unit ...
3votes
3answers
126views
Finding the vector potential
$$\nabla\times\mathbf{B}=\nabla\times\left(\nabla\times\mathbf{A}\right)=\nabla\left(\nabla\cdot\mathbf{A}\right)-\nabla^2\mathbf{A}=\mu_0\mathbf{J}\tag{5.62}$$ Whenever I try to work this out and ...
- 220
0votes
1answer
102views
What are some ways to derive $\left( \boldsymbol{E}\cdot \boldsymbol{E} \right) \nabla =\frac{1}{2}\nabla \boldsymbol{E}^2$?
For each of the two reference books the constant equations are as follows: $$ \boldsymbol{E}\times \left( \nabla \times \boldsymbol{E} \right) =-\left( \boldsymbol{E}\cdot \nabla \right) \boldsymbol{E}...
- 105
2votes
2answers
157views
How to calculate the rotation at a singularity?
An electrodynamics lecture asks me to prove that $$ \nabla \times \left( \frac{\vec{M} \times \vec{x}}{ |\vec{x}|^3} \right) = \frac{8 \pi}{3} \vec{M} \delta^3(\vec{x})- \frac{\vec{M}}{|\vec{x}|^3}+ \...
- 160
1vote
1answer
195views
Given a divergence free vector Field, how can I find the related vector potential without guesswork?
I am reading through Griffith's Electrodynamics 4e cover to cover, skipping no sections and doing all of the problems (sans the ones for masochists) as a passion project. I came to problem 1.53, in ...
